Invertible K.2/-local E-modules in C4-spectra

We compute the Picard group of the category of K.2/-local module spectra over the ring spectrum EhC4, where E is a height 2 Morava E-theory and C4 is a subgroup of the associated Morava stabilizer group. This group can be identified with the Picard group of K.2/-local E-modules in genuine C4 -spectra. We show that in addition to a cyclic subgroup of order 32 generated by E ^S1, the Picard group contains a subgroup of order 2 generated by E ^S7C, where is the sign representation of the group C4. In the process, we completely compute the RO.C4/-graded Mackey functor homotopy fixed point spectral sequence for the C4 -spectrum E.